Exponents and scientific notation
Exponents are shorthand for repeated multiplication. A handful of rules let you simplify powers quickly, and scientific notation uses powers of 10 to write very large and very small numbers.
What an exponent means
For a positive integer n, \(a^n\) means a multiplied by itself n times: \(2^5 = 32\). The number a is the base and n the exponent. Watch the base carefully: \(-3^2 = -9\), because only 3 is squared, while \((-3)^2 = 9\).
Product and quotient rules
When multiplying powers with the same base, add the exponents: \(x^3 \cdot x^4 = x^7\). When dividing, subtract them: \(\frac{x^9}{x^4} = x^5\). These rules work only when the bases match; \(2^3 \cdot 3^2\) cannot be combined into a single power.
Power rules
Raising a power to a power multiplies the exponents: \((x^3)^4 = x^{12}\). A power of a product spreads to each factor: \((2x)^3 = 8x^3\). A power of a quotient spreads to top and bottom: \(\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}\).
Zero and negative exponents
Any non-zero number to the power 0 equals 1: \(5^0 = 1\). A negative exponent means a reciprocal: \(x^{-n} = \frac{1}{x^n}\), so \(2^{-3} = \frac{1}{8}\). These definitions keep the product and quotient rules consistent: \(\frac{x^4}{x^4} = x^0 = 1\).
Simplifying with the rules
Combine the rules step by step. For example, \(\frac{(3x^2y)^2}{9x^5} = \frac{9x^4y^2}{9x^5} = x^{-1}y^2 = \frac{y^2}{x}\). Simplified answers usually avoid negative exponents.
Scientific notation
Scientific notation writes a number as \(a \times 10^n\), with \(1 \le |a| < 10\) and n an integer. Earth's average distance from the Sun, about 150,000,000 km, is \(1.5 \times 10^8\) km; a typical bacterium's length, about 0.000002 m, is \(2 \times 10^{-6}\) m. The exponent counts how many places the decimal point moves.
Calculating in scientific notation
To multiply, multiply the front numbers and add the exponents: \((3 \times 10^4)(2 \times 10^5) = 6 \times 10^9\). To divide, divide and subtract: \(\frac{8 \times 10^6}{4 \times 10^2} = 2 \times 10^4\). If the front number leaves the range 1 to 10, adjust: \(12 \times 10^5 = 1.2 \times 10^6\).
Worked example
Light travels about \(3 \times 10^8\) m/s, and the Sun is about \(1.5 \times 10^{11}\) m away. The travel time is \(\frac{1.5 \times 10^{11}}{3 \times 10^8} = 0.5 \times 10^3 = 5 \times 10^2\) seconds, or 500 s, a little over 8 minutes. Sunlight you see left the Sun about 8 minutes ago.
Key takeaways
An exponent counts repeated multiplication; check what the base is, since -3 squared is -9 but (-3) squared is 9. With the same base, multiplying adds exponents and dividing subtracts them; a power of a power multiplies exponents. Any non-zero number to the power 0 is 1, and negative exponents mean reciprocals. Scientific notation writes numbers as a times 10 to the n with a between 1 and 10; multiply by adding exponents and divide by subtracting them, then adjust the front number if needed.
Flashcard
The number being multiplied
Check your understanding
Simplify \(x^3 \cdot x^4\).